Supercoiling is ubiquitous in cellular DNA, which results from the topology of double-helical DNA [1]}, [2]}, [3]}, [4]}. For eukaryotic DNA, the large structural domains (\(\sim 1\)  M\(b\) ) emerge in the genome as the topologically associating domains (TADs)[5]}, [6]}, [7]}, which are further divided into smaller supercoiling domains(\(\sim 10^5 bp\) )[8]}, [9]}, [10]}. For the prokaryotic DNA, comparatively large structural domains and smaller supercoiling domains also present (e.g., \(E\) . \(coli\)  with an average domain size, \(\sim 10^{4}\)  \(bp\) ) [11]}, [9]}. Supercoils are generated in many crucial genetic processes and in turn regulates corresponding processes, including transcription, replication and genome condensation[13]}, [14]}, [15]}, [1]}, [17]}, [18]}, [19]}, [20]}. During transcription elongation, a RNA polymerase (RNAP) moves along the helical DNA and generates twists, which subsequently create positive supercoiling ahead of the RNAP or downstream along the transcription direction and negative supercoiling behind or upstream[13]}. The accumulation of the torsional stress can slow down or even stall the transcription elongation[14]}, [15]}.
Recent experiments show long-distance cooperative and antagonistic dynamics of multiple RNAPs via supercoiling[24]}, which justify the role of supercoiling as a long-range mediator[25]}. A previous modeling study pointed out that multiple RNAPs facilitate elongation by reducing collisions or traffic jam by torques[26]}; and another model suggested that multiple RNAPs also induce strong supercoiling to bring significant transcription fluctuations[27]}.
In addition, supercoiling contributes to the chromosome condensation[19]}, [20]}, during which the negative supercoils accumulate in the protein-free region of the DNA [30]}. Under tension, the accumulative supercoiling presents twisted and interwound coils called plectonemes, which may act as a driving force for compacting the chromatin fiber[30]}, [32]}.
